روشی جدید مبتنی بر الگوریتم های تکاملی و عددی برای حل معادلات لین-اِمدن
محورهای موضوعی : آنالیز عددی
سید رضا میرشفائی
1
,
هاشم صابری نجفی
2
,
اسماعیل خالقی
3
,
امیرحسین رفاهی شیخانی
4
1 - گروه ریاضی، واحد لاهیجان، دانشگاه آزاد اسلامی، لاهیجان، ایران
2 - گروه ریاضی، واحد لاهیجان، دانشگاه آزاد اسلامی، لاهیجان، ایران
3 - گروه مهندسی مکانیک، دانشکده مکانیک، دانشگاه گیلان، رشت، ایران
4 - گروه ریاضی، واحد لاهیجان، دانشگاه آزاد اسلامی، لاهیجان، ایران
کلید واژه: Numerical methods, Stellar structure, Genetic programming, Lane-Emden differential equations,
چکیده مقاله :
با پیشرفت کامپیوترها و توسعه صنعت تولید پردازنده ها، امروزه از الگوریتم های تکاملی نظیر شبکه های عصبی، الگوریتم ژنتیک و برنامه سازی ژنتیکی در حوزه های مختلف مهندسی استفاده می گردد. اما در حوزه ریاضیات کمتر از چنین روش هایی برای حل مسایل خاصی نظیر معادلات دیفرانسیل استفاده شده است. ایده موجود در این مطالعه ارائه روشی جدید و نوآورانه در حوزه ریاضیات کاربردی و اخترفیزیک است که مطابق با آن، بتوان روش های ریاضی را با روش های ابتکاری که پیش تر اکثراً در مسایل کاربردی و مهندسی از آن بهره گرفته می شد، تلفیق نموده و از آن برای حل معادلات دیفرانسیل مرتبه دوم غیرخطی لین-امدن که برآمده از فیزیک اجرام آسمانی و توصیف کننده ساختار ستاره ای و تئوری کره های گازی پلی تروپیک است، استفاده نمود. در این راستا یک روش ترکیبی جدید (GPNLE) مبتنی بر برنامه سازی ژنتیکی و روش عددی رونگه-کوتا برای تولید مدل های ریاضی با دقتی مطلوب از جواب معادلات لین- امدن معرفی شده است. صحت کارایی و انعطاف پذیری این روش ترکیبی بر اساس آزمایش های عددی انجام شده روی دسته هایی خاص و پرکاربرد از این نوع از معادلات مورد بررسی و قیاس با یک روش قدرتمند بر پایه چندجمله ای های چبیشف، قرار گرفته و نتایج مطلوبی برای نشان دادن اهداف مقاله حاصل شده است.
Evolutionary algorithms such as neural networks, genetic algorithms, and genetic programming are used in various engineering fields with the advancement of computers and the development of the processor manufacturing industry. But in mathematics, fewer than such methods have been used to solve special problems such as differential equations. This study aims to present a new and innovative method in applied mathematics and astrophysics, according to which mathematical methods can be combined with evolutionary processes, and it can be used to solve Lane-Emden nonlinear second-order differential equations. In this study, a novel GPNLE method based on an evolutionary algorithm (including genetic programming) and combining it with a numerical method (Runge-Kutta) is presented to generate mathematical models with appropriate accuracy from the solution of the second-order nonlinear Lane-Emden differential equations arising from astronomy. The present method's accuracy, efficiency, and flexibility based on numerical experiments performed on these equations have been analyzed and compared with a powerful method based on Chebyshev polynomials. The obtained result confirms the efficiency and validity of the presented method.
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